Does using baking soda instead of toothpaste damage your teeth?

Not really: Baking soda is a little abrasive but so is toothpaste. They both can aid in cleaning but the physical mechanical motion is the most important and the Fluoride in the toothpaste is second. I would rec toothpaste over baking soda.

Abrasive to teeth: Baking soda is abrasive to teeth so I do not recommend it. Any flouride toothpaste should be adequate if you are using the proper brushing techniques and floss.

Early toothpaste: Some parents use a little baking soda on infant teeth instead of the non-fluoride tooth paste when they have trouble finding it or to get medication stains off.Fluoride tp is not recommended early on when babies swallow a lot of the gel, i generally prefer to have a dentist apply the gel directly in early childhood & wait for regular Fluoride tp until the kid is ~3. I see little pbs with it in kids.

Yes: Use backing soda tooth paste which available and it is ada ( american dental association) approval.